https://docs.python.org/3/library/exceptions.html#SystemExit says: skip traceback, convert default None to 0, print any other non-int and convert to 1, and pass unprinted int to C exit().

I agree that IDLE should also print non-ints.  The IDLE doc should mention its non-exit behavior.

[https://docs.python.org/3/library/sys.html#sys.exit adds more about int return codes and that exit() only exits the process in the main thread.  Neither is relevant here.]

I believe the relevant code is the following, from run.Executive.runcode:

        except SystemExit:
            # Scripts that raise SystemExit should just
            # return to the interactive prompt
            pass
        except:
            self.usr_exc_info = sys.exc_info()
            if quitting:
                exit()
            # even print a user code SystemExit exception, continue
            print_exception()
            jit = self.rpchandler.console.getvar("<<toggle-jit-stack-viewer>>")
            if jit:
                self.rpchandler.interp.open_remote_stack_viewer()

The bare except clause, including the comment, is from 2003.  The 'except SystemExit' clause was added 2013 June 11 in #18196, a follow-up of comments in #5492.  The obsoleted comment should have been deleted.  The behavior suppressed was always printing traceback + message.  What should have been retained was printing non-int messages, but I don't think that either Roger or I were aware of that behavior.

One could argue that SystemExit in user code should trigger an exit from the user execution process, which would trigger a Shell Restart.  This would be closer to the standard behavior.  But it does not hurt, and may be better, to keep the process and let the user trigger a restart when wanted.  In the case that inspired #18196, the SystemExit came from site.py, not user code, and definitely should not cause a restart.

I will try something like the following:

        except SystemExit as e:
            ob = e.args[0]
            if not isinstance(ob, (type(None), int)):
                print('SystemExit: ' + str(ob), file=sys.stderr)

Since the message will be followed by a normal prompt rather than an exit, I want to 'enhance' the message with the prefix and error color.